Limit groups over coherent right-angled Artin groups

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

There is an important class of finitely presented groups known as automatic groups, in which many computations can be undertaken efficiently. In particular, there is a normal form for group elements which is the language of a finite state automation, and arbitrary words in the group generators can be reduced to normal form in quadratic time. The normal form can also be used to enumerate group elements, and to compute the (rational) growth function of the group. The theory of automatic groups is developed [2], and the algorithms are described in more detail in [4]. Implementations of the algorithms are available in a software package of the author ([3]).The word-hyperbolic groups form an important subclass of the automatic groups. They are defined by the property that geodesic triangles in the Cayley graph of the group are uniformly thin. In other words, there is a constant e, such a that any point on one of the sides of such a triangle is within distance e of the union of the other two sides of the triangle. Small cancellation groups and groups acting discretely and cocompactly on hyperbolic space are examples of word-hyperbolic groups. Their theory is developed in [1].The word problem in a word-hyperbolic group can be solved by a Dehn algorithm in linear time. Furthermore, Epstein has shown that the conjugacy problem can be solved in time O(n ln n). However, both of these algorithms require a prior knowledge of the constant e, and it does not seem to be easy to compute this constant in general.In practice, the quadratic time algorithm for the word problem coming from the automatic groups procedures performs very effectively, and it would be useful to have alternatives for the conjugacy problem.The automatic groups machinery does provide such an algorithm (which works in the the intermediate class of biautomatic groups), but that unfortunately is exponential.Alternative algorithms for conjugacy testing of elements and subgroups are currently being developed and implemented by a research student of mine, Joe Marshall. They have the minor disadvanatage that their basic versions work only in torsion-free groups, but adaptations to the more general case seem possible. They include also a test for malnormality of a subgroup H of G, which means that g-1Hg∩H = 1 for all g ∈ G\H. This test is important in topological applications.They make essential use of the boundary of the group. The points on the boundary can be defined as the equivalence classes of infinite geodesic rays starting from the base point of the Cayley graph, where two such rays are equivalent if they remain a bounded distance apart. The precise bound involved here turns out to be another universal constant d of the Cayley graph, but unlike e it can be computed in practice relatively easily. Roughly speaking, a certain linear function of d provides us with a bound on the length of the elements g that we need to consider as possible conjugating elements when testing for conjugacy or malnormality. To make the algorithm practical, we need to restrict this set of potential conjugators considerably more than this, and Marshall has come up with various tricks for doing this.

We discuss a number of problems in relatively hyperbolic groups. We show that the word problem and the conjugacy (search) problem are solvable in linear and quadratic time, respectively, for a relatively hyperbolic group, whenever the corresponding problem is solvable in linear and quadratic time in each parabolic subgroup. We also consider the class R of finitely generated toral relatively hyperbolic groups. We show that groups from R are commutative transitive and generalize a theorem proved by Baumslag in [Bau67] to R. Moreover, we discuss two definitions of (fully) residually-C groups, the classical Definition 1.2.1 and a modified Definition 1.2.5. Building upon results obtained by Ol'shanskii [Ol'93] and Osin [Osi10], we prove the equivalence of the two definitions for C = R. This is a generalization of the similar result obtained by Ol'shanskii for the class of torsion-free hyperbolic groups. Let R be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [KM12] that a finitely generated fully residually- group G embeds into an iterated extension of centralizers of . We deduce from their theorem that every finitely generated fully residually- group embeds into a group from R. On the other hand, we give an example of a finitely generated torsion-free fully residually-H group that does not embed into a group from R; H is the class of hyperbolic groups.

The three fundamental decision problems posed by Max Dehn in 1912 are the word problem, the conjugacy problem and the isomorphism problem. Let G be a group and let u and v be elements of G. The word problem asks for an algorithm for deciding whether u = v. The conjugacy problem asks about the existence of an element g ∈ G which conjugates u to v, i.e., v = g −1 ug. A solution of the conjugacy problem clearly contains a solution of the word problem. The isomorphism problem asks whether two group presentations define isomorphic groups. The word and conjugacy problems received much attention in the literature. For a summary of results concerning the conjugacy problem until 1987 see [13] and references therein.

We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan, and Kazachkov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over coherent RAAGs and toral relatively hyperbolic groups have a solvable word problem.

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

The conjugacy problem asks whether two words over generators of a fixed group G are conjugated, i.e., it is the problem to decide on input words x, y whether there exists z such that $$zx z^{-1} =y$$zxz-1=y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the conjugacy problem for two prominent groups: the Baumslag---Solitar group $$\mathbf{{BS}}_{1,2}$$BS1,2 and the Baumslag group $${\mathbf{{G}}}_{1,2}$$G1,2 (also known as Baumslag---Gersten group). The conjugacy problem in $${\mathbf{{BS}}}_{1,2}$$BS1,2 is complete for the circuit class $$\mathsf {TC}^0$$TC0. To the best of our knowledge $${\mathbf{{BS}}}_{1,2}$$BS1,2 is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group $${\mathbf{{G}}}_{1,2}$$G1,2 is an HNN-extension of $${\mathbf{{BS}}}_{1,2}$$BS1,2. Hence, decidability of the conjugacy problem in $$\mathbf{{G}}_{1,2}$$G1,2 outside the so-called black hole follows from Borovik et al. (Int J Algebra Comput 17(5/6):963---997, 2007). Decidability everywhere is due to Beese. Moreover, he showed exponential time for the set of elements which cannot be conjugated into $$\mathbf{{BS}}_{1,2}$$BS1,2 (Beese 2012). Here we improve Beese's result in two directions by showing that the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2 can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs, conjugacy in $${\mathbf{{G}}}_{1,2}$$G1,2 can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2 by reducing the divisibility problem in power circuits to the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2. The complexity of the divisibility problem in power circuits is an open and interesting problem in integer arithmetic. We conjecture that it cannot be solved in elementary time because we can show that it cannot be solved in elementary time by calculating modulo values in power circuits.

We prove that a countable group G can be embedded into a two-generator simple group S which is an amalgamated free product of groups G * F and F, where F and F are free groups on two generators.S is also the product of two commuting free subgroups.If G has solvable word problem, then we can construct a recursive presentation for S.1. Introduction.It is well known that every countable group is a subgroup of a simple two-generator group (see [3] and [7]).The main result of our paper is Theorem A. (a) A countable group G can be embedded into a simple two-generator group S which is an amalgamated free product of G * Fx and F, where F and Fx are both free groups on two generators.S is also the product of two commuting free subgroups.(b) If G has solvable word problem, we can construct the group S of (a) with a recursive presentation.It was first shown by R. J. Thompson [8] by a construction completely different from ours that a countable group with solvable word problem is a subgroup of a recursively presented finitely generated simple group.Our construction of the group S is similar to the construction of the simple free products of R. Camm [1], or see [6] for a slightly simplified version.2. The construction of the group S. Let 77 be a countable infinite group with an element x of order at least 5, and assume that the elements of 77 are numbered by the integers, 77 = {g" i G Z}, such that g0 = 1, gx = x, g2 = x2, g3 = x3, g4 = x4.Let A = (a) and F = (b,c) be free groups on a and b, c, respectively.Then the subgroup U of A * H generated by the elements of the form ", = aW *'eZ-{0}, where p is a permutation of Z -{0}, is a free group freely generated by the elements u" / g Z -{0}, with transversal AinA*H.Similarly, the elements Vj = bjcu\ 7GZ-{0), of F, where a is a second permutation of Z -{0}, freely generate a free subgroup V of F with transversal B = (b).

The decision problems in the title are solved using the solutions of the word and conjugacy problems obtained by Lyndon and Schupp, respectively, for certain classes of groups.

Stallings graphs for quasi-convex subgroups

We show that being finitely presentable and being finitely presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids. Our main tool is an elementary, but useful, geometric characterization of finite presentability for left cancellative monoids. We also give examples to show that this characterization does not extend to monoids in general, and indeed that properties such as solvable word problem are not isometry invariants for general monoids.

We study the monoid generated by certain Zariski-van Kampen generators in the positive homogeneous presented fundamental group of the complement of the logarithmic free divisor, called the type $\mathrm{B_{ii}}$ in the list by Sekiguchi. Although the monoid is cancellative, it turns out that the monoid is not Gaussian and, hence, is neither Garside nor Artin. Nevertheless, we show that the monoid carries certain particular elements similar to the fundamental elements in Artin monoid. Hence, we can solve the word problem and the conjugacy problem in the monoid and determine the center of it and the explicit form of the growth function for it. As a result, we can also solve the word problem and the conjugacy problem in the fundamental group, and determine the center of it (Theorem 5.8).

Word and conjugacy problems in groups with only a few defining relations

We prove that the limits of Baumslag–Solitar groups studied by the authors are non-linear hopfian C*-simple groups with infinitely many twisted conjugacy classes. We exhibit infinite presentations for these groups, classify them up to group isomorphism, describe their automorphisms and discuss the word and conjugacy problems. Finally, we prove that the set of these groups has non-zero Hausdorff dimension in the space of marked groups on two generators.

Algorithms detecting stability and Morseness for finitely generated groups